Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the derivative of a^x?

ln(a) a^x

(a^x)ln a

The derivative of the function $ a^x $, where $ a $ is a constant, can be derived using the properties of exponential functions and logarithms. When differentiating $ a^x $, it is helpful to express it in terms of the natural exponential function:

a^x = e^{x \ln(a)}

By applying the chain rule to the derivative of $ e^{u(x)} $ where $ u(x) = x \ln(a) $, we find:

\frac{d}{dx}(e^{u(x)}) = e^{u(x)} \cdot u'(x)

Calculating $ u'(x) $, we observe that:

u'(x) = \ln(a)

Now, substituting back into the differentiation formula gives us:

\frac{d}{dx}(a^x) = e^{x \ln(a)} \cdot \ln(a) = \ln(a) \cdot e^{x \ln(a)} = \ln(a) \cdot a^x

Thus, the derivative of $ a^x $ is \( \ln(a) \cdot a

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(a^x)/x

ln(x) a^x

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